Quick question about a hyperbola in $\log-\log$ space Let $u=\ln(x)$ and $v=\ln(y).$ Is it correct to conclude that $uv=1,$ in $\log-\log,$ space is a hyperbola, but in $x,y$ space it is not a hyperbola?

It is indeed correct to conclude that in log-log space $\ln(x)\ln(y)=1$ is the hyperbola $xy=1.$ Taking the natural logarithm of each point of the plot $\ln(x)\ln(y)=1$ yields the curve $xy=1.$